A Tutorial Introduction to the Logic of Parametric Probability

Aristotle's Logic Computed by Parametric Probability and Linear Optimization

A new computational method is presented to implement the system of deductive logic described by Aristotle in Prior Analytics. Each Aristotelian problem is interpreted as a parametric probability network in which the premises give constraints on probabilities relating the problem's categorical terms (major, minor, and middle). Each probability expression from this network is evaluated to yield a linear function of the parameters in the probability model. By this approach the constraints specified as premises translate into linear equalities and inequalities involving a few real-valued variables…

Parametric mixed sequent calculus

In this paper, we present a propositional sequent calculus containing disjoint copies of classical and intuitionistic logics. We prove a cut-elimination theorem and we establish a relation between this system and linear logic.

Parametric Presburger arithmetic: logic, combinatorics, and quasi-polynomial behavior

Parametric Presburger arithmetic concerns families of sets S_t in Z^d, for t in N, that are defined using addition, inequalities, constants in Z, Boolean operations, multiplication by t, and quantifiers on variables ranging over Z. That is, such families are defined using quantifiers and Boolean combinations of formulas of the form a(t) x <= b(t), where a(t) is in Z[t]^d, b(t) in Z[t]. A function g: N -> Z is a quasi-polynomial if there exists a period m and polynomials f_0, ..., f_{m-1} in Q[t] such that g(t)=f_i(t) for t congruent to i (mod m.) Recent results of Chen, Li, Sam; Calegari, Walk…

范畴证明理论导论

An Introduction to Categorical Proof Theory

本文是基于2021-2022年逻辑领域相关会议暑期学校教程扩充的范畴证明理论讲义，无需范畴论基础，仅需基础逻辑等知识，逐步引入范畴背景，探讨范畴方法在证明理论中的应用，侧重证明的哲学与非形式层面。

Polymorphism and the obstinate circularity of second order logic: a victims' tale

The investigations on higher-order type theories and on the related notion of parametric polymorphism constitute the technical counterpart of the old foundational problem of the circularity (or impredicativity) of second and higher order logic. However, the epistemological significance of such investigations, and of their often non trivial results, has not received much attention in the contemporary foundational debate. The results recalled in this paper suggest that the question of the circularity of second order logic cannot be reduced to the simple assessment of a vicious circle. Through a …